asymptotics of homoclinic bifurcation in a three-dimensional system

Nonlinear Dynamics21: 135–155, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

Asymptotics of Homoclinic Bifurcation in a Three-DimensionalSystem

M. BELHAQ and M. HOUSSNILaboratory of Mechanics, Faculty of Sciences Aïn Chock, Group of Nonlinear Oscillations and Chaos,BP 5366, Maârif, Casablanca, Morocco

E. FREIRE and A. J. RODRÍGUEZ–LUISDepartment of Applied Mathematics II, Escuela Superior Ingenieros, University of Sevilla, Camino de losDescubrimientos s/n, 41092 Sevilla, Spain

(Received: 22 October 1998; accepted: 19 May 1999)

Abstract. An analytical approach to predicting a critical parameter value of homoclinic bifurcation in a three-dimensional system is reported. The multiple scales method is first performed to construct a higher-order approx-imation of the periodic solution. A criterion based on a collision between the periodic orbit and the fixed pointinvolved in the bifurcation is applied. This criterion developed initially to predict homoclinic bifurcations in planarautonomous systems, is adapted here to derive a critical value of the homoclinic bifurcation in a specific three-dimensional system. To support our analytical predictions and to describe the dynamical behaviour of the system,a complete numerical study is provided.

Keywords: Three-dimensional systems, periodic orbit, multiple scales analysis, homoclinic bifurcation.

1. Introduction

The dynamics of three-dimensional systems near one of their periodic orbits is very richin terms of bifurcation and stability. Important behaviours include symmetry-breaking,period-doubling, Neimark–Sacker (bifurcation to invariant torus) and specially homoclinicor Shil’nikov bifurcation. See, for example, one of the pioneering works of Shil’nikov [29]and the works where a three-dimensional electronic circuit is studied by Freire et al. [15] andAlgaba et al. [1–3].

Indeed, homoclinic and heteroclinic orbits are of great importance from an applied pointof view. For instance, they form the profiles of travelling wave solutions in reaction-diffusionproblems (see, for instance, [13, 19]). Their existence can be a source of chaotic dynam-ics in three-dimensional systems (see, for example, [25, 29]). In static-dynamics analogies,a homoclinic orbit corresponds to a spatially localized post-buckling state [30]. The well-known Chua circuit governed by a three-dimensional differential equation system is now oneof the most studied systems because of its richness and simplicity [10]. Khibnik et al. [18]analyzed Chua’s circuit equations with a smooth nonlinearity and reported the central role ofhomoclinicity in the model. Nekorkin and Kazantsev [24] investigated the travelling wavesin a one-dimensional circular array of Chua’s circuits. It was shown that the problem can bereduced to an analysis of the periodic orbit of a three-dimensional system of ordinary dif-ferential equations describing the individual dynamics of Chua’s circuit. Within this context,

136 M. Belhaq et al.

the development of new analytical methodologies to predict bifurcations in three-dimensionalsystems is an exciting issue and strongly required.

The investigation of homoclinic bifurcation has been receiving much attention from boththe analytical and numerical points of view (see, for example, [11, 22, 28]). The classicalmathematically rigorous approach to predicting bifurcations of homoclinic orbits is Mel-nikov’s method. This approach mainly requires the distance between the manifolds of theperturbed system to vanish. Recently, another analytical method to predict the homoclinicbifurcations in autonomous self-excited two-dimensional systems was reported in [5]. Thisapproach formally approximating the infinite period of the bifurcating periodic orbits. Moreprecisely, the condition considered at such bifurcations is the limit when the period goes toinfinity or the vanishing of the frequency of the periodic solution. In [31], a semi-analyticaland numerical process was developed to determine the separatrices and limit cycles of stronglytwo-dimensional nonlinear oscillators. Conditions under which a limit cycle is created ordestroyed were derived. In [4] and [6], a formal analytical criterion to predict homoclinicbifurcations in autonomous two-dimensional systems was reported. This criterion is mainlybased on the collision, at the homoclinic bifurcation, of the periodic orbit with the equilibriuminvolved in the bifurcation. Mathematically speaking, this criterion is equivalent to the Mel-nikov method (for details, see [7]). Note, however, that the collision criterion is accessible viaapproximations of periodic orbits. The Melnikov approach, on the other hand, circumventsperiodic orbits by aiming directly at the separatrices.

In this paper, we apply the collision criterion to formally derive an approximate homoclinicbifurcation exhibited by the three-dimensional system

x = µx − y − xz,y = µy + x,z = −z + x2z+ y2. (1)

System (1) may be thought of as a response control system consisting of a damped linearoscillator inx, y variables and a control variablez. One should note that this oscillator hasnegative damping for positiveµ. The ‘dot’ denotes the time derivative,x, y andz are scalarvariables andµ is a scalar parameter. For system (1), symmetry-breaking and period-doublingbifurcations has been analytically investigated by Rand [26], Nayfeh and Balachandran [21],Belhaq and co-workers [8, 9].

The approximate homoclinic bifurcation is provided by a suitable adaptation to system (1)the criterion proposed in [4, 6] for autonomous planar systems.

The origin (x0 = 0,y0 = 0,z0 = 0) of Equations (1) is an equilibrium, stable forµ < 0 andunstable forµ > 0 so that a Hopf bifurcation occurs atµ = 0. As the parameterµ increasesfrom zero, the periodic orbit undergoes a symmetry-breaking bifurcation atµ = µSB1. Asµincreases again, this orbit becomes unstable and a new stable periodic orbit of twice the periodappears by period-doubling atµ = µPD2,1. This orbit undergoes a second period-doublingbifurcation atµ = µPD4,1. A detailed numerical study is given in Section 4. A heteroclinicconnection between the nontrivial equilibria organizes the branch of the principal periodicorbit. Homoclinic connections of double, quadruple, octuple, etc., pulse also act as organizingcenters of the dynamics. The presence of cascades of period-doubling bifurcations, and thenof chaotic attractors, is also pointed out.

Using the center manifold theory and a near-identity transformation, Rand and co-workers[26, 27] constructed a first-order approximation of the limit cycle near the Hopf bifurcation.

Homoclinic Bifurcation in a Three-Dimensional System137

The critical valueµPD2,1 (≈ 0.45), corresponding to the first period-doubling bifurcation wasapproached by studying the stability of the orbit. Nayfeh and Balachandran [21] used themethod of multiple scales [20] to obtain, as in [26], the same first-order approximation of theperiodic solution. The critical valuesµSB1 (≈ 0.30), corresponding to the symmetry-breakingbifurcation, andµPD2,1 (≈ 0.4405) were numerically approximated using Floquet theory [23].

In [9] a higher-order approximation of the periodic orbit, using a higher-order multiple-scales expansion, was constructed. This expansion was successfully used to predict thebifurcation valuesµSB1 = 0.31 andµPD2,1 = 0.446. Recently, Belhaq et al. [8] derivedan analytical approximation of the critical valueµPD4,1(≈ 0.486) corresponding to the secondperiod-doubling.

In this work, we specifically present an analytical scheme based on formal asymptoticexpansions to approximate the parameter value at which a homoclinic bifurcation takes placein such a system. We first derive a higher-order asymptotic expansion of the periodic orbitby using the multiple scales method. The collision criterion of homoclinicity [6] is thenapplied and adapted to the system under study in order to predict a homoclinic bifurcation.Comparisons to numerical results are provided for validating our analytical prediction.

This paper is organized as follows. In Section 2, a higher-order approximation of the peri-odic solution following Hopf bifurcation is obtained using the multiple-scales method to ahigher-order. Section 3 is devoted to the prediction of the homoclinic connection. A detailednumerical study of system (1), which provides comparisons to the analytical approach as wellas the full dynamical behaviour of the system, is given in Section 4. Finally, we present someconclusions in Section 5.

2. Asymptotic Expansion

To determine an approximation of the periodic orbit of system (1), we apply the method ofmultiple scales [20], by seeking a uniformal valid expansion of the form

x =6∑n=1

εnxn(T0, T1, T2, T3, T4, T5)+ · · · ,

y =6∑n=1

εnyn(T0, T1, T2, T3, T4, T5)+ · · · ,

z =6∑n=1

εnzn(T0, T1, T2, T3, T4, T5)+ · · · , (2)

whereTn = εnt are the time scales andε is a small positive dimensionless parameter whichis the order of the amplitude of the motion. This parameter is artificially introduced to serveas a bookkeeping device, in obtaining the approximate solution, and will disappear naturally,since in the expansion (9), for example, the termεa always appears as a block. Hence,following Equations (6), we see that this term can be written in the original parameter form:εa = (2/3)√10ε2µ2 = (2/3)√10µ. For this reason, one usually setsε equal to unity in thefinal analysis [22, 23]. The control parameter is expanded asµ = ε2µ2+O(ε3). Substitutingthis last relation and Equations (2) into Equations (1), taking into account thatx0 = 0, y0 = 0,z0 = 0 and equating coefficients of like powers ofε, we obtain, at different orders ofε, the


following systems of successive approximationsxn, yn, zn:

o(ε1):

D0x1 + y1 = 0,

D0y1 − x1 = 0,

D0z1+ z1 = 0. (3)

o(εi, i ≥ 2):

D0xi + yi = µ2xi−2 −i∑

j=0

xjzi−j −i∑

j=1

Djxi−j ,

D0yi − xi = µ2yi−2 −i∑

j=1

Djyi−j ,

D0zi + zi =i∑

j=0

yjyi−j −i∑

j=1

Djzi−j +i∑

j=0

(xi−j

j∑k=0

zkxj−k

), (4)

whereDn = ∂/∂Tn. The explicit expression to higher orders of system (4) are detailed in [9].Using Equations (3) and (4) fori = 2,3, the solution up to second order is given by [21]

x(t) = −εa sinθ +O(ε3),

y(t) = εa cosθ +O(ε3),

z(t) = 1

2ε2a2

[1+ 1

5cos(2θ)+ 2

5sin(2θ)

]+O(ε3), (5)

where

a = 2

3

√10µ2, θ =

(1− ε

2a2

20

)t +O(ε3). (6)

These values ofa andθ are obtained from the two conditions

D1A = 0, 2D2A− 2µ2A+ 9+ 2i

5A2A = 0, (7)

that make the secular terms in Equations (4) vanish fori = 2 andi = 3. Now let us investigatea higher-order approximation of the limit cycle. The general solution of system (4) fori = 3is given by

x3 = 3

40(2i − 1)A3 e3iT0 + cc,

y3 = 2+ i40

A3 e3iT0 − i(9+ 2i)

10A2AeiT0 + cc,

z3 = 0, (8)


where ‘cc’ stands for the complex conjugate of the preceding expressions. Hence, from Equa-tions (5–8), the uniformally valid expansion to third order of the periodic solution is givenby

x(t) = −εa sinθ − ε3

(3a3

160cos(3θ)+ 3a3

80sin(3θ)

)+O(ε4),

y(t) = εa cosθ + ε3

(a3

20cosθ + 9a3

40sinθ + a

3

80cos(3θ)− a3

160sin(3θ)

)+O(ε4),

z(t) = ε2a2

2

(1+ 1

5cos(2θ)+ 2

5sin(2θ)

)+O(ε4), (9)

wherea andθ are given by Equations (6).On the other hand, systems (4) fori = 4,5,6, allows one to determine the approximation

of the periodic orbit up to fifth order. Indeed, the elimination of the secular terms in system (4)for i = 4 leads to the condition

D3A = 0 (10)

and then the solution up to fourth order can be written as

x4 = 0,

y4 = 0,

z4 = 9i − 2

20(1+ 4i)A4 e4iT0 − 2

(1+ 2i)2AD2Ae2iT0

− 22+ 51i

20(1+ 2i)A3Ae2iT0 + 11− 7i

5(AA)2−D2(AA)+ cc. (11)

The equation that eliminates the secular terms in Equations (4) fori = 5 is given by thecondition

D4A = µ2KA2A+HA3(A)2, (12)

where

K = 38

25+ 61

100i, H = −3052+ 861i

2000. (13)

Similarly, from the higher-order system we obtain the last condition that cause the secularterms in Equation (4) to vanish fori = 6

D5A = 0. (14)

Therefore, the set of five conditions to be resolved is given by

D1A = 0, (15)

2D2A− 2µ2A+ 9+ 2i

5A2A = 0, (16)

D3A = 0, (17)


D4A = µ2

(38

25+ 61

100i

)A2A− 3052+ 861i

2000A3(A)2, (18)

D5A = 0. (19)

SubstitutingA = (1/2)a eiβ (wherea andβ are real quantities) into Equations (15–19),separating the real and imaginary parts, we obtain

D2a = µ2a − 9

40a3, (20)

D4a = 19

50µ2a

3 − 763

8000a5, (21)

D4β = − 1

20a2, (22)

D4β = 61

400µ2a

2− 861

32000a4. (23)

The solution of Equation (20) is

a = C(T4)

√µ2− 9

40a2 exp(µ2T2), (24)

wherea andµ2 satisfy the condition

a2 <40

9µ2. (25)

Solving Equation (24) fora yields

a = C(T4)√µ2 exp(µ2T2)√

1+ 940(C(T4))2 exp(2µ2T2)

, (26)

whereC(T4) is an arbitrary function ofT4. Now the substitution of Equation (26) intoEquation (21) provides an ordinary differential equation onC(T4) in the form

1+ 940(C(T4))

2 exp(2µ2T2)

1950(C(T4))3− 79

8000(C(T4))5 exp(2µ2T2)dC = µ2

2 exp(2µ2T2)dT4, (27)

in whichT2 is to be treated as a constant. This may be integrated to give

(C(T4))2

[exp

(− σ

(C(T4))2α

)+ E exp

(FT4 + c

α

)]= 19

50exp

(FT4+ c

α

), (28)

where

E = 79

8000exp(2µ2T2), F = 2µ2

2

αexp(2µ2T2),

σ = 50

19, α = 763

3040σ exp(2µ2T2) (29)

andc denotes a constant of integration. Since this last equation is a transcendental one forC(T4), one cannot obtain a closed form expression forC(T4) to be substituted into the


previous expression (26) fora. Nevertheless, in order to overcome this difficulty, it turnsout that an asymptotic approximation forC(T4) can be obtained by expanding the functionexp(−σ/(C(T4))

2α) and retaining only the two first terms of the expansion. Expanding, sub-stituting these terms in Equation (28) and resolving inC(T4), we obtain an approximation ofC(T4) as follows

C(T4) =[σα+ 19

50 exp(FT4+ c

α

)1+ E exp

(FT4+ c

α

) ]1/2

. (30)

Hence the previous expression ofa, given by Equation (26), becomes

a =[µ2(

3040763 exp(−2µ2T2)+ 19

50 exp(2µ2T2+ 11552

3815µ22T4+ c

α

))]1/2[1447763 + 763

8000exp(2µ2T2+ 11552

3815µ22T4+ c

α

)]1/2 . (31)

It follows that the new approximation of the amplitude of the limit cycle isa →20√(38/3815)µ2 as t → ∞. This expression verifies, as expected, the condition given by

Equation (25). Note that the first approximation of the amplitudea is given by Equation (6).Consequently, from Equations (9) and (11), and the solution of Equations (4) fori = 5, weobtain the following fifth-order approximation of the periodic solution

x(t) = −εa sinθ − ε3

(3a3

160cos(3θ)+ 3a3

80sin(3θ)

)+ ε5

(− 32a5

15360cos(5θ)+ a5

15360sin(5θ)+ 11587a5

486400cos(3θ)− 3247a5

972800sin(3θ)

)+O(ε6), (32)

y(t) = εa cosθ + ε3

(a3

20cosθ + 9a3

40sinθ − a3

160sin(3θ)+ a

3

80cos(3θ)

)+ε5

(−197771a5

4620800cosθ + 1289a5

30400sinθ + 186181a5

110899200cos(3θ)

− 180177a5

166348800sin(3θ)− a5

76800cos(5θ)− 32a5

76800sin(5θ)

)+O(ε6), (33)

z(t) = ε2a2

2

(1+ 1

5cos(2θ)+ 2

5sin(2θ)

)+ ε

4a4

80

(757

38+ 161

190sin(2θ)− 2187

190cos(2θ)+ cos(4θ)− 1

2sin(4θ)

)+O(ε6), (34)

wherea andθ are now given by the new approximations

a = 20

√38

3815µ2, θ =

(1− 1

20ε2a2 + 2765

243200ε4a4

)t +O(ε6). (35)


(a)

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-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

YN

FF

I

T

T

I

(b)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5X

-1.5

-1.0

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0.0

0.5

1.0

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Y

I

I

T

T

F FN

Figure 1. Comparison of different approximations of periodic orbits for: (a)µ = 0.3; (b) µ = 0.4. LabelNcorresponds to the exact orbit obtained by numerical integration.I indicates first-order approximation,T denotesthe third-order approximation orbit andF corresponds to the orbit obtained with the fifth-order approximation.

In Figure 1 we compare the approximations at different orders of the periodic orbit obtained bythe multiple-scales technique, Equations (32–35), with the periodic orbit obtained numericallyby integrating system (1) for two parameter values ofµ.

3. Homoclinic Double-Pulse Connection

This section is concerned with an analytical prediction of a homoclinic connection occurringin system (1). The strategy followed consists of two steps. First, we formally construct ahigher-order approximation of the bifurcating periodic orbit forµ > 0. Then the criteriongiven in [4, 6], for the planar autonomous systems, is adapted to system (1) and applied topredict an approximate critical value of this homoclinic bifurcation. This criterion is based on


PD2,1

PD2,2

PD4,1PD4,2

PD4,4

PD4,5

PD4,6

S1,1

S1,2

S1,3

S1,4

S2,1

S2,2

PD4,3

▼▼

▼▼■

■

●

● ●

●

●

●

●

●

●

■

■

■

Period

µ

SB1SB2

SB3 SB4

Hopf

● ■

HetHom Hom2,1 2,2

PD8,2PD8,1

Figure 2. Partial bifurcation diagram of the periodic orbit emerged from the Hopf bifurcation. In this qualitativefigure, the solid line means stable periodic orbit and the dashed line means saddle periodic orbit. We have usedthe following convention: empty square for the Hopf bifurcation, inverted triangle for the symmetry-breakingbifurcation, filled circle for the period-doubling bifurcation and filled square for the saddle-node bifurcation. ThenotationPDi,j means that from that period-doubling bifurcation point, a branch of periodic orbits emerges with aperiod approximatelyi times the period of the principal branch (we will call it aniT -orbit). The labelj indicatesthat it is thej th point of the kind we have mentioned in our description of the complex bifurcation diagramexhibited by system (1). Analogously, we use the notationSi,j to indicate thej th saddle-node bifurcation foundin a branch ofiT -orbits. We denote byHomi,j thej th i-pulse homoclinic connection of the nontrivial equilibria.Note that we represent the period of theT , 2T , 4T , 8T -orbits divided, respectively, by 1, 2, 4 and 8.

the idea that, at the homoclinic bifurcation, there exists a timet for which the periodic orbitand the involved equilibrium collide.

The stationary solutions of system (1), corresponding tox = y = z = 0, arex = y = z =0 which is an unstable focus, and the two equilibria

x+ = −µy+y+ =

√zs

1+µ2zs

zs = 1+µ2

µ

,

x− = −µy−y− = −

√zs

1+µ2zs

zs = 1+µ2

µ

. (36)

If 0 < µ < µc ≈ 0.24, then the fixed points(x±, y±, zs) are saddles, whereas they aresaddle-focussed ifµ > µc.

An analysis of the eigenvalues of the fixed points(x±, y±, zs) shows that the variationof the periodic orbit in the direction of the control variablez is the most important whencompared to the variation of the amplitude along the other directions. In consequence, we canapply, as a first approach, the criterion of homoclinic bifurcation only in thez-direction. Then,the criterion reduces to the condition

z(t) = zs, (37)

where z(t) is the approximation of the periodic orbit (see the Appendix) andzs is thecoordinate of the fixed point given in Equations (36).

We point out that the approximate periodic solution up to fifth order (Equations (32–35))has the same shape as the one obtained numerically which is lying in the vicinity of theHom2,2


branch (see Figure 2). This observation suggests us using this analytical approximation to in-vestigate the corresponding homoclinic bifurcation. The approximate solution of the periodicorbit in thez-direction at different orders inε is given in the Appendix.

Using the approximation of the periodic orbit inz (seez(3) in the Appendix) up to thirdorder, the criterion (37) leads to the condition

−1+ 5

3µ2 = 0. (38)

Resolving this last equation, we obtain a first approximation of the homoclinic bifurcation,namelyµHB1 = 0.774.

Clearly, this approximation is not in agreement with the numerical calculation (µHB =0.540, see the next section). To improve this analytical prediction, we shall consider the fifth-order approximation of the periodic solution inz (seez(5) in the Appendix). Similarly, usingthis last approximation, criterion (37) now leads to the equation

−1+ 1061

763µ2 + 1087104

582169µ3 = 0. (39)

Resolving Equation (39) gives a second approximation of the homoclinic bifurcationµHB2 =0.625.

This result is still not good enough, but comes closer to the numerical result. This stim-ulating observation tells us to go further in our calculation and to construct a higher-orderapproximation of the periodic solution.

Following the same procedure as above and using now the seventh-order approximation inz (seez(7) in the Appendix), we obtain the following new condition

−1+ 1061

763µ2 + 1087104

582169µ3+ 5149190570912

2993556660675µ4 = 0. (40)

Hence, resolving Equation (40) leads to the third approximation of the homoclinic bifurcationµHB3 = 0.574. At this stage, this result can be considered as a good prediction compared tothe result obtained by the numerical simulation given below.

It is clear that to investigate the higher-order approximationsµHB4, µHB5, . . ., in orderto improve the approximation of the critical value of homoclinicity, the hand computationsbecome very cumbersome to perform. However, we can use the same idea which allowed usto predict a very good approximation of the critical value corresponding to symmetry-breakingbifurcation [9]. It turns out that combining the above three values forµHB1, µHB2 andµHB3,we can conjecture the following result

µHB1− µHB2∼= 0.15, µHB2− µHB3

∼= 0.05= 0.15

3. (41)

Assuming now that this process continues for the other successive approximations ofµHB, wecan construct the following trigonometric series

µHBn = µHB1− α[

1− (13

)n−1

1− 13

], (42)

whereα = 0.15. The limit of this series asn goes to infinity provides the following criticalvalue of the homoclinic bifurcation

µHB = limn−→∞µHBn = 0.545. (43)


This conjecture has improved the approximation of the homoclinicity, but it is clear that itcan be justified only by performing further calculations, for instance approximateµHB4 andverify whether or notµHB4−µHB3 follows the process given by Equation (41). The numericalstudy performed in the next section gives the valueµ(Hom2,2) = 0.540.

4. Numerical Study

In this section, we describe the dynamical behaviour found in system (1). To do this, we haveused the software continuation code AUTO94 [12] as well as DSTOOL [17]. Particularly, thecritical valuesµSB1, µPD2,1, µPD4,1 andµHB mentioned in the previous sections, correspond,respectively, to the bifurcation pointsSB1, PD2,1, PD4,1 andHom2,2 described below.

Note that this system has the same symmetry the Lorenz equations have: it is invariant tothe change(x, y, z) −→ (−x,−y, z). The origin is always an equilibrium point, and twoother equilibria are given by Equations (36). The origin is stable forµ < 0 and it exhibitsa Hopf bifurcation forµ = 0 and then becomes a saddle-focus forµ > 0. The nontrivialequilibria are always saddle (two negative eigenvalues) in the region of interest (µ > 0).

The stability analysis of the Hopf bifurcation (see, for instance, [14]) reveals that it issupercritical: a stable symmetric periodic orbit emerges forµ > 0. The evolution of this peri-odic orbit is schematized in the bifurcation diagram of Figure 2. In this qualitative figure, wehave indicated the Hopf bifurcation by an empty square, the symmetry-breaking bifurcationby an inverted triangle, the period-doubling bifurcation by a filled circle, and the saddle-node bifurcation by a filled square. The notationPDi,j means that, from that period-doublingbifurcation point, a branch of periodic orbits with period approximatelyi-times the period ofthe principal branch (we will call it aniT -orbit) emerges. The labelj indicates that it is thej th point of the kind we have mentioned in our description of the complex bifurcation diagramexhibited by system (1). Analogously, we use the notationSi,j to indicate thej th saddle-nodebifurcation found in a branch ofiT -orbits. We denote byHomi,j the j th i-pulse homoclinicconnection of the nontrivial equilibria. Note that we represent the period of theT , 2T , 4T ,8T -orbits divided, respectively, by 1, 2, 4 and 8.

First, the periodic orbit exhibits a symmetry-breaking bifurcation,SB1 (µ = 0.3150232),to become a saddle orbit and a pair of asymmetric stable periodic orbits emerges. The principalorbit recovers its stability in a second pitchfork bifurcation,SB2 (µ = 0.6939566). This stablesymmetric orbit collapses in a saddle-node bifurcation (fold),S1,1 (µ = 0.721608), with asaddle symmetric orbit, exhibiting a new saddle-node bifurcationS1,2 (µ = 0.6334804) wherea new stable symmetric orbit appears. This orbit undergoes a symmetry-breaking bifurcation,SB3 (µ = 0.6391011) to become unstable and a new pair of asymmetric stable periodic orbitsemerges. The saddle symmetric orbit is stable again when it exhibits the pitchfork bifurcationSB4 (µ = 0.6625362). The interval of the parameterµ where this orbit is stable is narrowas it suffers a saddle-node bifurcation,S1,3 (µ = 0.6646371). The saddle symmetric orbitundergoes a new saddle-node bifurcation,S1,4 (µ = 0.6641073). Finally, the resulting stablesymmetric orbit approaches a heteroclinic connection between the non-trivial equilibria,Het(µ = 0.6658963).

The behaviour of the orbit that emerged from the Hopf bifurcation corresponds to the typ-ical wiggle of a periodic orbit around a homoclinic/heteroclinic connection (see, for instance,[16]). Because of the symmetry the system has, symmetry-breaking and saddle-node bifurca-tions are combined in the principal branch in the way we have just described. In Figures 3a and


(a) (b)

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0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

(c) (d)

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Y

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Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 3. Phase portraits of the heteroclinic orbit (µ = 0.6658963) ((a) and (b)) and the two saddle symmetricperiodic orbits that coexist with this global connection ((c) and (d)).

3b, we show the projection of the heteroclinic orbitHet on the(x, y)-plane and on the(x, z)-plane, respectively. The two saddle symmetric periodic orbits that coexist with this globalconnection are drawn in Figures 3c and 3d.

Now we focus on the pair of asymmetric stable periodic orbits that emerged atSB1. Theseorbits become saddle when they exhibit a period-doubling bifurcation,PD2,1 (µ = 0.4403559).Note that this flip bifurcation was analytically predicted to occur forµPD1= 0.446 by Belhaqand Houssni [9]. From such a bifurcation point, a stable periodic orbit of approximately twicethe period of the original orbit emerges. The stability of the pair is recovered in a new flipbifurcation PD2,2 (µ = 0.6815652) and this branch disappears atSB2. In short, there is abranch of asymmetric periodic orbits connectingSB1 and SB2 where two period-doublingbifurcations occur.

The asymmetric 2T -orbit (in fact a pair, due to the symmetry the system has) born atPD2,1

becomes non-stable in a flip bifurcationPD4,1 (µ = 0.4765392) where a 4T -orbit emerges(this period-doubling bifurcation was analytically predicted to occur forµPD2 = 0.486, in[8]). It becomes stable again in a new period-doubling bifurcationPD4,2 (µ = 0.4942767)and finally approaches a double-pulse homoclinic connectionHom2,1 (µ = 0.503992). In thissituation there is not a Feigenbaum period-doubling cascade, since the 4T -orbit that emergesatPD4,1 always becomes stable and its branch disappears atPD4,2.


(a) (b)

X

Y

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00-1.5

-1.0

-0.5

0.0

0.5

1.0

X

Y

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00-1.5

-1.0

-0.5

0.0

0.5

1.0

(c) (d)

X

Y

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00-1.5

-1.0

-0.5

0.0

0.5

1.0

X

Y

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00-1.5

-1.0

-0.5

0.0

0.5

1.0

Figure 4. (a) Asymmetric periodic orbit, that emerged fromSB1, just at the point where it exhibits the flip bi-furcationPD2,1 (µ = 0.4403559). (b) 2T -orbit at the pointPD4,1 (µ = 0.4765392). (c) Double-pulse homoclinicconnectionHom2,1 (µ = 0.503992). (d) A stable 4T -orbit forµ = 0.485.

In Figure 4a, we show the asymmetric periodic orbit that emerged fromSB1, just at thepoint where it exhibits the flip bifurcationPD2,1. In Figure 4b we have represented the 2T -orbit at the pointPD4,1. The double pulse homoclinic connectionHom2,1 is sketched in Figure4c. A stable 4T -orbit is showed in Figure 4d, forµ = 0.485.

On the other hand, the asymmetric 2T -orbit that emerged fromPD2,2 becomes non-stablein a period-doubling bifurcationPD4,3 (µ = 0.6788854) where a 4T -orbit emerges. It be-comes stable again in a new flip bifurcationPD4,4 (µ = 0.5344726) but quickly undergoes afold S2,1 (µ = 0.5338808). The new saddle 2T -orbit collapses, in a saddle-node bifurcationS2,2 (µ = 0.5440001), with a stable periodic orbit. In the way it approaches the homoclinicconnectionHom2,2 (µ = 0.5404654), it exhibits a pair of period-doubling bifurcations,PD4,5

(µ = 0.5434981) andPD4,6 (µ = 0.5414545). Note that these last two points are connectedwith a branch of 4T -orbits that exhibits two flip bifurcationsPD8,1 (µ = 0.5430199) andPD8,2

(µ = 0.5427911). Finally, the 8T -orbits that emerged from such points remain stable. Hereagain, the Feigenbaum cascade is not present.

We are now interested in the branches that emerge from the pointsSB3, SB4, PD4,3 andPD4,4.We have sketched them in separate figures for the sake of clarity, due to the rich bifurcationdiagrams these orbits exhibit.


PD2,3

PD2,4

PD2,5

PD4,7

PD4,9PD

2,6

PD4,10PD8,3

PD16,1

PD 32,1

PD8,4

PD4,8

PD16,2PD 32,2

Hom2,4Hom2,3Period

µ

▼

▼●

●

●

●

●

●●

●●

●

●●

●

●●

●

●

●

●

●●

●

●

●

●

■■

■

■

■■

SB3

SB4

●

Figure 5. Partial bifurcation diagram that shows the branches that emerged from pointsSB3 andSB4.

First, we consider the branch of asymmetric periodic orbits connecting the pointsSB3 andSB4 (see Figure 5). These orbits exhibit up to four flip bifurcations,PD2,3 (µ = 0.6423311),PD2,4 (µ = 0.6477739),PD2,5 (µ = 0.6538292) andPD2,6 (µ = 0.6611240).

The first two points are connected by a branch of 2T -orbits which undergoes a pair of flipbifurcationsPD4,7 andPD4,8. The branch of 4T -orbits that emerged from these two points alsopresents two period-doubling bifurcationsPD8,3 andPD8,4. In fact, we find two Feigenbaumcascades of period-doubling bifurcations that are connected for the corresponding branchesof 2nT -orbits (n = 1,2, . . .). See in Figure 6 the first subharmonic periodic orbits (all aresaddle).

However, the branches of 2T -orbits that emerge fromPD2,5 andPD2,6 are disconnected.The branch started atPD2,5 wiggles around a double-pulse homoclinic connectionHom2,3

(µ = 0.6559152). In fact, a new Feigenbaum cascade of period-doubling bifurcations ap-pears in such a way that the branches emerged from the period-doubling points close toPD2,5

approach homoclinic orbits of quadruple-, octuple-, etc., pulses. See the first homoclinic orbitsin Figure 7.

The branch started atPD2,6 also wiggles around a double-pulse homoclinic connectionHom2,4 (µ = 0.6576122), but there are no other homoclinic connections of quadruple-,octuple-, etc., pulses. We can see how the period-doubling bifurcation pointsPD4,9 andPD4,10

are connected via a branch with two saddle-node points (s-shaped). This shape recursivelyoccurs in the branches joining the corresponding flip points. The presence of cusp bifurcationsof periodic orbits would be detected if a second parameter of the system was moved.

However, the connection between the flip points in the upper part of this branch that endsatHom2,4 occurs with branches without saddle-node bifurcations.

The bifurcations present in the branches started at the flip pointsPD4,3 and PD4,4 aresketched, respectively, in Figures 8 and 9. In the first case we observe a cascade of period-doubling bifurcations (PD4,3, PD8,5, PD16,3, etc.) whose branches wiggle (exhibiting flip andsaddle-node bifurcations), respectively, around the homoclinic connections of quadruple pulseHom4,1, octuple pulseHom8,1, 16-pulseHom16,1, etc.

In Figure 9, similar wiggles occur around the homoclinic connections of quadruple pulseHom4,2, octuple pulseHom8,2, 16-pulseHom16,2, etc. In Figure 10a, we show a saddle periodic or-


(a) (b)

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(c) (d)

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 6. Periodic orbits coexisting forµ = 0.645 with a chaotic attractor, in the vicinity of two Feigenbaumcascades that appear between the bifurcation pointsPD2,3 and PD2,4: (a) 2T -orbit; (b) 4T -orbit; (c) 8T -orbit;(d) 16T -orbit.

bit on the branch emerged fromPD2,2 just before exhibitingPD4,4. The homoclinic connectionsHom4,2, Hom8,2 andHom16,2 are drawn in Figures 10b–10d.

We finish this numerical study by showing in Figure 11 two of the chaotic attractors exhib-ited by system (1) as a consequence of the presence of period-doubling cascades. The chaoticattractor that appears after the period-doubling cascadePD4,4, PD8,6, PD16,4, etc. (see Figures 2and 9) is shown forµ = 0.53476 in Figure 11a. To investigate its structure we have taken asectionz = 1.5 and obtained an apparently one-dimensional intersection that is represented inFigure 11b. This fact points out the strong contraction the system exhibits for this value of theparameter. On the other hand, two projections of the chaotic attractor that exists forµ = 0.645between the two cascades started, respectively, atPD2,3 andPD2,4 (see Figure 5) are drawn inFigures 11c–11d. A detailed study of these attractors is left for future work.

5. Conclusions

In this paper we have analytically approximated, using a formal asymptotic expansion, acritical value corresponding to homoclinic bifurcation in a specific three-dimensional system.To obtain this critical value, we have applied the criterion, proposed initially by Belhaq [4]for the autonomous planar systems, to the three-dimensional one considered in the present


(a) (b)

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(c)

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Figure 7. Homoclinic orbits of: (a) double pulseHom2,3 (µ = 0.6559152); (b) quadruple pulse (µ = 0.6551305);(c) octuple pulse (µ = 0.6550226).

Period

µ

PD4,3

Hom4,1 Hom8,1 Hom16,1

PD8,5

PD16,3

●

●

●

●

●

●

●●

●●

● ●

●

●

●●

●●

■

■

■

■

■■

■

■

■

■

■

■■

■

■

■

■

■

●

Figure 8. Partial bifurcation diagram that show the branches emerged from pointPD4,3.


Period

µ■

■●

●■

■

■●

■●

●

●

●

●

●

■ ■

●

●

■

■

■●

■

●

PD8,6

PD4,4

PD16,4

Hom16,2 Hom8,2 Hom4,2

Figure 9. Partial bifurcation diagram that shows the branches that emerged from pointPD4,4.

(a) (b)

X

Y

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00-1.5

-1.0

-0.5

0.0

0.5

1.0

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0-1.5

-1.0

-0.5

0.0

0.5

1.0

(c) (d)

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0-1.5

-1.0

-0.5

0.0

0.5

1.0

X

Y

-1.5 -1.0 -0.5 0.0 0.5 1.0-1.5

-1.0

-0.5

0.0

0.5

1.0

Figure 10. (a) Saddle 2T periodic orbit, forµ = 0.53476, in the branch that emerged fromPD2,2 just beforeexhibiting PD4,4. (b) Homoclinic connection of quadruple pulseHom4,2 for µ = 0.5485049. (c) Homoclinicconnection of octuple pulseHom8,2 for µ = 0.5480355. (d) Homoclinic connection of sixteen-pulseHom16,2for µ = 0.5480346.


(a) (b)

x

y

x

y

(c) (d)

x

y

x

z

Figure 11. Chaotic attractor that exists forµ = 0.53476: (a) projection onto thex-y plane; (b) intersection withthe planez = 1.5. Chaotic attractor present forµ = 0.645: (c) projection onto thex-y plane; (d) projection ontothex-z plane. Crosses denote saddle equilibria.

work. The strategy followed to derive such an asymptotic approximation of the homoclinicbifurcation was motivated by two facts. First, the criterion used here, based on the collisionat the homoclinic bifurcation between the bifurcating periodic orbit and the equilibrium, hasgiven a very good prediction of critical values in the two-dimensional autonomous systemsusing trigonometric functions in the perturbation method. In a recent work [7], it has beenshown that this criterion combined formally with the Jacobian elliptic functions is, math-ematically speaking, equivalent to the Melnikov method. The second motivation regards theconjecture established in [9]. This conjecture consisted in deriving three approximations ofthe critical value of the symmetry-breaking bifurcation of the system under study using thethree first successive approximations of the periodic orbit, respectively. These three values arecombined in a suitable manner, as in the present work, to construct a geometrical series. In the


limit, the series provides a good approximation of the critical value of the symmetry-breakingbifurcation. As a natural step, the collision criterion was adapted to the three-dimensionalsystem and the conjecture was applied in the same way to construct the critical value of thehomoclinic bifurcation after deriving the three first successive approximations of the homo-clinic bifurcation. To obtain these three critical values, we have formally approximated theperiodic orbit using a higher-order multiple scales expansion.

The adaptation of the collision criterion to three-dimensional systems provides a goodprediction of the homoclinic bifurcation even under a certain assumption for which the cri-terion was applied only in thez-direction. An open and interesting problem is to show howthe present three-dimensional analysis may be improved by combining the collision criterionwith the Jacobian elliptic functions as was done in two-dimensional systems [7]. To supportour analytical approaches and to describe the dynamic behaviour of the system, we haveperformed numerical simulations using software continuation codes.

With our paper, we hope to have achieved a first small step in suggesting a new alternativeanalytical scheme to predict homoclinic connections in three-dimensional systems.

Acknowledgements

This work has been carried out through the cooperation between Morocco and theJunta deAndalucía. The hospitality of the University of Sevilla is acknowledged. We also thank F.Fernández-Sánchez for help in drawing several figures and F. J. Muñoz-Almaraz for helpfulcomments on the use of LaTeX.

Appendix

∗ orderO(ε3)

z(3)(t) = a2

2

(1+ 1

5cos 2θ + 2

5sin 2θ

)+O(ε4).

a = 2

3

√10µ and θ =

(1− a

2

20

)t +O(ε4).

∗ orderO(ε5)

z(5)(t) = a2

2

(1+ 1

5cos 2θ + 2

5sin 2θ

)+ 757

3040a4

+ a4

(161

15200sin 2θ − 2187

15200cos 2θ + 1

80cos 4θ − 1

160sin 4θ

)+O(ε6).

a = 20

√38

3815µ and θ =

(1− 1

20a2+ 2765

243200a4

)t +O(ε6).

∗ orderO(ε7)

z(7)(t) = a2

2

(1+ 1

5cos 2θ + 2

5sin 2θ

)+ 757

3040a4+ 38301

51200a6 − 49

16µa4 + 2µ2a2


+ a4

(161

15200sin 2θ − 2187

15200cos 2θ + 1

80cos 4θ − 1

160sin 4θ

)+ a6

(559

5683200cos 6θ − 3281

2841600sin 6θ + 1567

816000cos 4θ + 3523

272000sin 4θ

)− a6

(151869

640000cos 2θ + 54579

320000sin 2θ

)− 2µ2a2

(11

125cos 2θ + 2

125sin 2θ

)+ µa4

(27397

40000cos 2θ + 15004

40000sin 2θ − 407

27200cos 4θ − 574

27200sin 4θ

)+O(ε8).

wherea andθ are still given as in orderO(ε5).

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asymptotics of homoclinic bifurcation in a three-dimensional system

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